We evaluate the performance of models for the covariance structure of stock returns, focusing on their use for optimal portfolio selection. We compare the models' forecasts of future covariances and the optimized portfolios' out-of-sample performance. A few factors capture the general covariance structure. Portfolio optimization helps for risk control, and a three-factor model is adequate for selecting the minimum-variance portfolio. Under a tracking error volatility criterion, which is widely used in practice, larger differences emerge across the models. In general more factors are necessary when the objective is to minimize tracking error volatility.

[We evaluate the performance of models for the covariance structure of stock returns, focusing on their use for optimal portfolio selection. We compare the models' forecasts of future covariances and the optimized portfolios' out-of-sample performance. A few factors capture the general covariance structure. Portfolio optimization helps for risk control, and a three-factor model is adequate for selecting the minimum-variance portfolio. Under a tracking error volatility criterion, which is widely used in practice, larger differences emerge across the models. In general more factors are necessary when the objective is to minimize tracking error volatility.]